Theorem cocanfo 37706

Description: Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Assertion

Ref	Expression
cocanfo	⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐺 = 𝐻)

Proof of Theorem cocanfo

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 769	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹))
2	1	fveq1d 6909	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = ((𝐻 ∘ 𝐹)‘𝑦))
3		simpl1 1190	. . . . . . 7 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐹:𝐴–onto→𝐵)
4		fof 6821	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
5	3, 4	syl 17	. . . . . 6 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐹:𝐴⟶𝐵)
6		fvco3 7008	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦)))
7	5, 6	sylan 580	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦)))
8		fvco3 7008	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
9	5, 8	sylan 580	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
10	2, 7, 9	3eqtr3d 2783	. . . 4 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)))
11	10	ralrimiva 3144	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → ∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)))
12		fveq2 6907	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐺‘(𝐹‘𝑦)) = (𝐺‘𝑥))
13		fveq2 6907	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐻‘(𝐹‘𝑦)) = (𝐻‘𝑥))
14	12, 13	eqeq12d 2751	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 → ((𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ (𝐺‘𝑥) = (𝐻‘𝑥)))
15	14	cbvfo 7309	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
16	3, 15	syl 17	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → (∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
17	11, 16	mpbid 232	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥))
18		eqfnfv 7051	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
19	18	3adant1 1129	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
20	19	adantr 480	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
21	17, 20	mpbird 257	1 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐺 = 𝐻)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ∘ ccom 5693   Fn wfn 6558  ⟶wf 6559  –onto→wfo 6561  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571

This theorem is referenced by: (None)